How can I instruct draftwatermark package to exclude the first few pages

You can change a boolean to suppress the printing:

\documentclass{article}
\usepackage{draftwatermark}
\usepackage{blindtext}

\begin{document}
\makeatletter\@sc@wm@stampfalse\makeatother
\blindtext[1]

\newpage 
\makeatletter\@sc@wm@stamptrue\makeatother
\blinddocument

\end{document}

This seems to work. Set the lightness to 1 for the pages on which you do not want to have a watermark, and set it to a different value for the other pages.

\documentclass{article}
\usepackage{draftwatermark}
\usepackage{lipsum}

\begin{document}
\SetWatermarkLightness{1}
\lipsum[1-10]
\clearpage
\SetWatermarkLightness{0.5}
\lipsum[10-20]

\end{document}

The same result is obtained with

\documentclass{article}
\usepackage{draftwatermark}
\usepackage{lipsum}

\begin{document}
\SetWatermarkLightness{\ifnum\thepage<3
1
\else
0.5
\fi}
\lipsum[1-10]
\clearpage
\lipsum[10-20]

\end{document}

with which you can switch on the watermark from p. 3 onward.